Efficient weight function for semiparametric weighted NLS — morie_nls_weight

\(W(x) = 1/\sigma^2(x)\) attains the single-index efficiency bound \(\Omega_{SI} = \{E[1(X \in A_x)\sigma^{-2}(X)\,\partial G\,\partial G']\}^{-1}\) (2.32).

Usage

morie_nls_weight_function(
  x,
  y,
  bandwidth = NULL,
  weights = NULL,
  beta_hat = NULL
)

Arguments

x: numeric matrix of covariates.
y: numeric response.
bandwidth: bandwidth; Silverman's rule when NULL.
weights: supplied weights; efficient ones estimated when NULL.
beta_hat: index direction; first canonical direction when NULL.

Value

list: beta, weights, sigma2_hat, omega, omega_SI, C, D, max_weight, efficient_weight_used, efficiency_loss_from_unknown_G, rate_loss_from_unknown_G, bandwidth, n, d, method.

Details

Not knowing sigma^2 costs nothing. It can be replaced by a consistent estimate and the bound is still attained, by the two-step procedure the section gives: fit with \(W = 1\), then regress the SQUARED residuals nonparametrically on X and refit with \(W = 1/s_n^2\).

Not knowing G does cost something. Except in special cases \(\Omega_{SI}\) exceeds the bound achievable with G known. The price of a nonparametric link is a loss of EFFICIENCY, not of rate, and the two are returned separately because conflating them is the usual error.

The covariance is the sandwich \(C_n^{-1}D_nC_n^{-1}\) with \(D_n\) carrying \(W^2\), not \(W\): only \(W^2\) makes the sandwich collapse to \(\Omega_{SI}\) at \(W = 1/\sigma^2\), which is what the book asserts the efficient weight achieves. Mirrors morie.fn.hrzwfun.

References

Horowitz, Sec. 2.5.2, eq. (2.32); Ichimura (1993), Chamberlain (1986), Newey and Stoker (1993).

Examples

set.seed(1)
x <- cbind(rnorm(100), rnorm(100))
y <- tanh(x %*% c(1, -0.6)) + rnorm(100) * 0.3
morie_nls_weight_function(x, y, beta_hat = c(1, -0.6))$efficient_weight_used
#> [1] TRUE